Error Monte Carlo
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that we use for the average. A possible measure of the error monte carlo error analysis is the ``variance'' defined by: (269) where and The monte carlo standard error ``standard deviation'' is . However, we should expect that the error decreases with monte carlo error estimation the number of points , and the quantity defines by (271) does not. Hence, this cannot be a good measure of the error. monte carlo simulation error Imagine that we perform several measurements of the integral, each of them yielding a result . Thes values have been obtained with different sequences of random numbers. According to the central limit theorem, these values whould be normally dstributed around a mean . Suppouse that
Monte Carlo Standard Deviation
we have a set of of such measurements . A convenient measure of the differences of these measurements is the ``standard deviation of the means'' : (270) where and Although gives us an estimate of the actual error, making additional meaurements is not practical. instead, it can be proven that (271) This relation becomes exact in the limit of a very large number of measurements. Note that this expression implies that the error decreases withthe squere root of the number of trials, meaning that if we want to reduce the error by a factor 10, we need 100 times more points for the average. Subsections Exercise 10.1: One dimensional integration Exercise 10.2: Importance of randomness Next: Exercise 10.1: One dimensional Up: Monte Carlo integration Previous: Simple Monte Carlo integration Adrian E. Feiguin 2009-11-04
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Monte Carlo Error Calculation
Author ManuscriptsPMC3337209 Am Stat. Author manuscript; available in PMC 2012 monte carlo integration error Apr 25.Published in final edited form as:Am Stat. 2009 May 1; 63(2): 155–162. doi: 10.1198/tast.2009.0030PMCID: PMC3337209NIHMSID: NIHMS272824On monte carlo method integration the Assessment of Monte Carlo Error in Simulation-Based Statistical AnalysesElizabeth Koehler, Biostatistician, Elizabeth Brown, Assistant Professor, and Sebastien J.-P. A. Haneuse, Associate Scientific InvestigatorElizabeth Koehler, Department http://www.northeastern.edu/afeiguin/phys5870/phys5870/node71.html of Biostatistics, Vanderbilt University, Nashville, TN 37232;Contributor Information.Elizabeth Koehler: ude.tlibrednav@relheok.e; Elizabeth Brown: ude.notgnihsaw@bazile; Sebastien J.-P. A. Haneuse: gro.chg@s.esuenah Author information ► Copyright and License information ►Copyright notice and DisclaimerSee other articles in PMC that cite the published article.AbstractStatistical experiments, more commonly referred to as Monte Carlo or simulation studies, are used to study the https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3337209/ behavior of statistical methods and measures under controlled situations. Whereas recent computing and methodological advances have permitted increased efficiency in the simulation process, known as variance reduction, such experiments remain limited by their finite nature and hence are subject to uncertainty; when a simulation is run more than once, different results are obtained. However, virtually no emphasis has been placed on reporting the uncertainty, referred to here as Monte Carlo error, associated with simulation results in the published literature, or on justifying the number of replications used. These deserve broader consideration. Here we present a series of simple and practical methods for estimating Monte Carlo error as well as determining the number of replications required to achieve a desired level of accuracy. The issues and methods are demonstrated with two simple examples, one evaluating operating characteristics of the maximum likelihood estimator for the parameters in logistic regression and the other in the context of using the bootstrap to obtain 9
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